CCLASSIFYING RATIONAL PARALLELEPIPEDS
RANDALL L. RATHBUN
Abstract.
It is proposed that the name ‘Diophantine parallelepiped’ be applied to rational parallelepipeds.By examining the count of possible right angles at any vertice, the parallelepipeds become classified into5 types: acute anorthgonal triclinic ; obtuse anorthgonal triclinic ; monorthogonal bi-clinic ; bi-orthogonalmonoclinic ; and rectangular cuboid ; There is sufficient confusion in the present literature to warrant anattempt at clarity for these types of pipeds and to offer a classification scheme.The possible rational components of the Diophantine parallelepiped are examined and some results fromcomputer searches are also presented. Introduction to Diophantine Parallelepipeds
Parallelepipeds have been examined, in Diophantine analysis of Number Theory, but a lack of clarity stillseems to exist.We define a rational parallelepiped in n -dimensions as a polytope spanned by n vectors (cid:126)v , . . . , (cid:126)v n in avector space over the rationals, Q , or integers, Z span( (cid:126)v , . . . , (cid:126)v n ) = t (cid:126)v + · · · + t n (cid:126)v n for t i ∈ Q for i = 1 → n Here we are interested in three dimensions, n = 3, so a rational parallelepiped is a prism determined by the 3rational basis vectors (cid:126)a , (cid:126)b , (cid:126)c . The prism has 8 vertices, 3 pairs of parallel faces which are all parallelograms,and 12 edges (3 distinct) ∈ Q .(1) span( (cid:126)a , (cid:126)b , (cid:126)c ) = t (cid:126)a + t (cid:126)b + t (cid:126)c for t i =1 → ∈ Q abc Figure 1.
The rational parallelepiped with 3 rational basis vectors (cid:126)a , (cid:126)b , and (cid:126)c In Figure 1, the lengths of the basis vectors, (cid:126)a , (cid:126)b , and (cid:126)c , are rational ∈ Q , and they determine theparallelepiped. 2. Classifying the Parallelepipeds by orthogonality
Literature abounds with examples confusing rectangular parallelepipeds with monoclinic pipeds. Thebi-clinic or mon-orthogonal case is hardly even known. The rectangular parallelepiped is called the ‘ cuboid ’, Date : November 27, 2018.2010
Mathematics Subject Classification.
Key words and phrases. integer cuboid, rectangular cuboid, perfect parallelepiped. a r X i v : . [ m a t h . N T ] N ov RANDALL L. RATHBUN or ‘ brick ’, or ‘
Euler brick ’, or even the ‘ orthotope ’, engendering confusion over the preferred name for thesame object.In an attempt at clarity for Diophantine analysis, this author proposes a classification scheme basedupon the maximum count of right angles which might occur at any of the 8 vertices in the Diophantineparallelepiped.Consider the 3 surface angles from Figure 1., α between (cid:126)a and (cid:126)b ; β between (cid:126)a and (cid:126)c ; and γ between (cid:126)b and (cid:126)c . There may be none, or 1, or 2, or 3 right angles at a vertice of the Diophantine parallelepiped. All 8vertices need to be considered.3. Five Classes of Rational Diophantine Parallelepipeds
There are 5 unique classes of parallelepipeds which do exist, depending upon the count of right angles atthe origin [the intersection of the basis vectors], as shown and proved in Appendix A.
Table 1.
The five classes of rational parallelepipeds by surface angles at a vertice.Name Comment Vertice Group(s)acute (anorthic) [no ] [-1 -1 1] [ 1 1 1] obtuse (anorthic) [no ] [-1 -1 -1] [-1 1 1] [-1 -1 0] [-1 0 1] [ 0 1 1] [-1 0 0] [ 0 0 1] [ 0 0 0]
In the table above, the classsifier for the vertice group is shown in the Vertice Group(s) column. Themeaning is as follows: 1 = acute angle; 0 = right angle; and − • the Diophantine acute anorthic parallelepiped (acute triclinic) • the Diophantine obtuse anorthic parallelepiped (obtuse triclinic) • the Diophantine 1-ortho parallelepiped (biclinic) • the Diophantine 2-ortho parallelepiped (monoclinic) • the Diophantine 3-ortho parallelepiped (rectangular) Please note that for the first two classes, the acute anorthic , and the obtuse anorthic , that no right angleexists at any vertex of the parallelepiped.The labeling of the pipeds using the terms anorthic, or triclinic, the relatively unknown biclinic and thewell known monoclinic is deliberate, and comes from the mineral crystallography classification scheme, sincethe crystals systems share some of the same morphology as the Diophantine pipeds.Thus the proposed classification scheme then depends upon whether or not (0,1,2,3) right angles existfor at least one of the eight vertices of the piped (actually at the origin or intersection of the basis vectorscreating the piped). 4.
Rational Parallelepipeds and their components
We need to take a careful look at the rational components of the Diophantine piped for Diophantineanalysis. Due to the extensive symmetry and parallelism of the parallelepiped figure, many of its componentsare congruent, thus insuring rationality for all elements in the same congruent group, such as the edges, theface diagonals, the skew triangles, the face or body triangles, and the parallelogram areas.
LASSIFYING RATIONAL PARALLELEPIPEDS 3
Table 2.
Diophantine parallelepiped classification scheme into 5 classes. proposed name acute obtuse 1-ortho 2-ortho rectangular common name triclinic triclinic parallelepiped monoclinic cuboidright angle(s)
Derivation from 3 rational or integer basis-vectors.
Again, as in eq(1) we first start with 3 finitevectors which span R as shown in figure 2. Let their magnitudes be rational ∈ Q or preferably as integer ∈ ( Z ). ~v = a~v = b~v = c Figure 2.
The three basis vectors, (cid:126)v , (cid:126)v (cid:126)v .with magnitudes a , b , c ∈ Z Call the integer lengths of these vectors a , b , and c respectively. Using these 3 vectors we can create 3unique parallelograms, using a pair of the 3 vectors, and then combine the 3 parallelograms into a paral-lelepiped.Where 3 parallelograms exist, with their sides of length [ a, b ], and [ a, c ], and [ b, c ], then a parallelepipedsolution exists, where a , b , and c are the magnitudes of the 3 basis vectors. We label the 3 pairs of thediagonals of the 3 parallelograms as d,e ; f,g ; and h,j .parallelogram sides diagonals1 a,b d,e a,c f,g b,c h,j Table 3. a, b, c for 3 parallelograms with matching sides4.2.
The Parallelogram equation and rational parallelograms.
The parallelogram equation (2) showsthat we can pick 3 values of a parallelogram to be rational ∈ Q .(2) 2( a + b ) = c + d where a, b are the sides and c, d are the diagonals of the parallelogram. We can pick 3 rational values fora parallelogram, satisfying equation (2), say sides a, b and the diagonal c , then the other diagonal d mayor may not be rational. If the second diagonal is rational, we call the parallelogram the rational or integerparallelogram (upon scaling).It is noted that these integer parallelograms are created from integer triangles, so every parallelogramhas at least 1 rational diagonal, and if a parallelepiped is constructed from 3 such triangles, then at least 3diagonals, say d , f , and h , from Table 3 are rational, while the remaining three e , g , and j may or may notbe rational. RANDALL L. RATHBUN
Locating coordinates in R for permuted tetrahedrons of a piped. Using these 3 parallelograms,which have at least 1 rational diagonal each, 48 tetrahedrons can be created from the permutations, as 6sets of 8 tetrahedrons. However only 1 set of the 8 tetrahedrons is necessary to completely specify all 6 sets.A convenient and unique collection covering the 8 possible arrangements of a convenient set is shown inTable 4 below.
Table 4.
Combinations of the sides/diagonals of 3 Parallelograms for a TetrahedronTetrahedron Sides a d b f c h a d b f c j a d b g c h a d b g c j a e b f c h a e b f c j a e b g c h a e b g c jDepending upon whether or not the second diagonal(s) is rational or not, we can create an integertetrahedron , in at least 1 case of the 8 given above.The reason we deal with integer tetrahedrons is that they provide us with the 6 key lengths: a , b , c , d , e , f ; which enable us to fix the vertices of the Parallelepiped into R Cartesian space.An integer tetrahedron is shown below in figure 3 with 6 integer (or rational) sides, a , b , c , d , e , & f .These sides are slightly relabeled from the set of 8 previously given so that the equations (3, 4) below givingthe lattice locations are correct.You can quickly recognize the 3 vectors (cid:126)v , (cid:126)v and (cid:126)v creating edges a , c , and e in the figure: a bc de f Figure 3.
Integer tetrahedron.We have to locate the vertices of this tetrahedron, given the lengths, into R , and I choose the followingmethod to fix the vertices v , v , v , and v into Cartesian 3D space. We find the 3 vertices first.(3) let tetrahedron edges be a, b, c, d, e, f ∈ Zv = [0 , , v = [ a, , B ) = a + c − b ac and sin( B ) = (cid:112) ( a + b + c )( a + b − c )( a + c − b )( b + c − a )2 acv = c [cos( B ) , sin( B ) , LASSIFYING RATIONAL PARALLELEPIPEDS 5
By using the intersections of three S spheres from the 3 known vertices [0 , , v , and v with the fourthvertice v , that vertice can be located.(4) x = a + e − d ay = c + e − f − rx s where r = c · cos( B )and s = c · sin( B ) both from v then z = ± (cid:112) e − x − y v = [ x, y, z ]The + z value was used (although − z is another solution).Most of the time, the vertices are located in a quadratic field K = Q ( √ D ), occasionally they can be ona rational lattice.We have to follow the same process in equations (3),(4) for all 8 integer tetrahedrons in Table 4. Theparallelepiped (1 of 8) is derived from these 3 vectors and we strictly label the vertices as shown and followthis order for the rest of this paper.1 23 45 67 8 b bb bb bb b Figure 4.
The labeled Diophantine parallelepiped.This labeled parallelepiped contains several (possibly rational) subcomponents, and they are examined inthe following §§ .4.4. The 28 Diophantine Piped Diagonals.
The first subcomponent to be considered are the diagonals,a line segment between 2 vertices. There are 28 diagonals in the Diophantine piped, the parallelogram inwhich they reside is given also, if it exists.
Table 5.
28 Diagonals in the Diophantine piped with parallelograms12 Face Diagonals1-4 2-3 5-8 6-7 F1234 F56781-6 2-5 3-8 4-7 F1256 F34781-7 3-5 2-8 4-6 F1357 F24684 Body Diagonals1-8 3-6 2-7 4-5 B1368 B2457B1278 B3456B1458 B2367
RANDALL L. RATHBUN
12 edges1-2 3-4 5-6 7-8 F1234 F56781-3 2-4 5-7 6-8 F1234 F56781-5 2-6 3-7 4-8 F1256 F3478There are 6 face parallelograms containing 12 diagonals, and 6 body parallelograms with 4 unique diago-nals, since they share the same set of 4 diagonals in a permutated manner.4.5.
The 56 Diophantine Piped Triangles.
Triangles are composed of 3 connected diagonals, or any 3vertices. There are 56 triangles to consider in the Diophantine piped. However only 48 reside on the face orbody parallelograms.
Table 6.
48 Diophantine Piped Triangles on Parallelograms. (cid:52) (cid:52) (cid:52) (cid:52)
123 234 F1234 567 678 F5678124 134 F1234 568 578 F5678125 256 F1256 347 478 F3478126 156 F1256 378 348 F3478135 357 F1357 246 468 F2468137 157 F1357 248 268 F2468127 278 B1278 345 456 B3456128 178 B1278 346 356 B3456136 368 B1368 245 457 B2457138 168 B1368 247 257 B2457148 158 B1458 236 367 B2367145 458 B1458 237 267 B2367There are also 8 triangles unaccounted for. These are not found on the parallelogram components of theDiophantine piped. I call them the ‘skew triangles’ as they are not located on any of the 12 parallelograms.The 8 skew non-parallelogram triangles are
Table 7. (cid:52) (cid:52) (cid:52) (cid:52)
146 147 167 467853 852 832 532The matching congruent triangles listed above are given in a column, preserving the isometry of thelengths of any two vertices of the triangle, following the order of the vertice labeling.4.6.
The 12 Diophantine Piped Parallelograms.
The Diophantine piped also contains 12 parallelo-grams.
LASSIFYING RATIONAL PARALLELEPIPEDS 7
Table 8.
12 Parallelograms in the Piped.face parallelogramsF1234 F1357 F1256F5678 F2468 F3478body parallelogramsB1278 B1368 B1458 B2367 B2457 B3456The face parallelograms given above in a column in Table 8. are congruent, and have the same area anddiagonal lengths. However please note that the 6 body parallelograms have distinct areas.4.7.
The Diophantine Parallelepiped Itself.
Finally there is the parallelepiped itself, which has 8 or-dered vertices such that there is a pair of 4 vertices which are co-planar and parallel. This figure is conven-tionally called the prism.There is one component to consider for rationality, the volume:parallelepiped volumeP123456784.8.
Summarizing the Diophantine Analysis for all 83 Components.
Considering all the componentsand the duplications of lengths which occur in the piped, there is a unique signature which can be checkedfor rationality:
Rational Signature Considerations • • • •
56 triangle areas* (48 subsumed in parallelograms, 8 are not) • • • • a , b , c , d , e , and f for the integer tetrahedron edges actually led to degenerate ‘flat’tetrahedrons where all three vectors (cid:126)v , (cid:126)v , (cid:126)v were co-planer, or spanning only R .Putting all this together results in the unique rational check signature for a given Diophantine paral-lelepiped.4.9. Unique Rationality Check of the Diophantine Piped.
We discover that not all 83 componentsneed to be checked for rationality. Since there are congruences between the lengths of the edges and diagonals,and congruences between the areas of the triangles, face parallelograms, and body parallelograms then notall need to be checked. In fact:
RANDALL L. RATHBUN •
27 components need to be checked for rationality
We provide an example of this signature for three sample pipeds, where 0 denotes an irrational value and 1a rational value.
Table 9.
Three sample Diophantine parallelepipeds
Basis Vectors Tetrahedron Sides v v v a b c d e f(103,0,0) ( − , √ − , − √ , √ √ − √ √
35) 10 27 23 26 24 41(44,0,0) (0,117,0) (0,0,240) 44 125 117 244 240 267
Table 10.
27 rational component checks for the three sample Diophantine pipeds edges skew triangles face diagonals body diagonals face area body area vol1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 01 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 01 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 Rational constraints for Diophantine objects
There is an geometric increase in the rationality requirements as the embedding dimension of the geometricobject increases. For this reason, it is necessary to examine both rational triangles and parallelograms, beforeconsidering
Diophantine parallelepipeds , in order to better understand those requirements.5.1.
Rational Triangles.
In Diophantine analysis of Number Theory, we can study triangles, for example,for rational components. Since the triangle is 3 sided polygon, (it is actually determined by 2 non-linearvectors), the following properties need to be checked for rationality.
Triangle Constraints • rational sides (3 sides) • rational area (1 area)This means that 2 = 16 possibilities exist, when considering the rationality of the triangle. But col-lectively all 3 sides are considered rational and we definitely are interested if the area is rational. In facttriangles with this property have been given a special name: ‘Heron triangles’ .We won’t consider internal parts of the triangle, such as the medians, or the angle bisectors, or the perpen-dicular bisectors, or other civians, for example, although their study in Diophantine analysis is important.We are simply making the point here that we have 2 components to consider for rationality: the sides andthe area .If we force the triangle to have 3 integer sides, a, b, c ∈ Q , then we only have to consider the area ∈ Q .The answer to the question, can we find a triangle with integer sides and rational area? has been knownin antiquity, the 3,4,5 right triangle with area 6 is the classic answer.Less well known is the fact that the primitive 5,5,6 isosceles Heron triangle, and the primitive 5,5,8isosceles Heron triangle, both with area 12, are the next smallest integer triangles with rational area. Infigure 5 below, note that the 5-5-6 Heron triangle was created from joining two 3-4-5 right triangles alongthe side length 4.The next 2 triangles are the composite 6,8,10 right triangle with area 24, and the primitive Heron triangle9,10,17 with area 36 which is not isosceles. It should be pointed out that if the triangle sides ∈ Z , and the LASSIFYING RATIONAL PARALLELEPIPEDS 9
34 5area = 6 65 5area = 12
Figure 5.
The 3,4,5 Right Triangle and 5,5,6 isosceles Heron Triangletriangle has rational area, then its area ∈ Z also. If the sides are fractional ∈ Q , and if the area is rational,then the area ∈ Q .Of note is the Heron formula for area of a triangles with sides a , b , and c (5) A = 14 (cid:112) ( a + b + c )( − a + b + c )( a − b + c )( a + b − c )which is used to help discover rational area for the triangle in question. These triangles can be parameterized,see § Rational Parallelograms.
We now consider parallelograms. They are 4 sided planar figures, with2 pairs of congruent parallel edges; ( a , c ), a = c and a (cid:107) c and ( b , d ), b = d and b (cid:107) d . They also can bedetermined by two non-linear vectors but there are 2 additional considerations to be added to the Diophantineanalysis.The parallelogram has the following properties to check for rationality. Parallelogram Constraints • rational sides (4 sides) • rational diagonals (2 diagonals) • rational area (1 area)This means that we have to consider any one of the 2 = 128 permutations of the parallelograms propertiesfor rationality. Let the parallelogram have sides ( a , c ), ( b , d ), with diagonals d , d and area A . Consideringthe pairs of sides as a unit, we then have only 32 (2 ) combinations to consider. But forcing both pairs ofsides to be rational further shortens down the 128 possibilities to just 8 cases. Table 11.
Possible rationality checks for a parallelogramparallelogram component ( a, c ) ( b, d ) d d A case 1 rational ? (cid:88) (cid:88) – – –case 2 rational ? (cid:88) (cid:88) (cid:88) – –case 3 rational ? (cid:88) (cid:88) – (cid:88) –case 4 rational ? (cid:88) (cid:88) (cid:88) (cid:88) –case 5 rational ? (cid:88) (cid:88) – – (cid:88) case 6 rational ? (cid:88) (cid:88) (cid:88) – (cid:88) case 7 rational ? (cid:88) (cid:88) – (cid:88) (cid:88) case 8 rational ? (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) From examination, it can be seen that case 3 is identical to case 2 and case 7 is identical to case 6, becauseorder is not important, so this narrows our possibilities to just 6 actual cases:
Table 12.
Actual rationality checks for the 6 cases of a parallelogramparallelogram component ( a, c ) ( b, d ) d d A case 1 rational ? (cid:88) (cid:88) – – –case 2 rational ? (cid:88) (cid:88) (cid:88) – –case 3 rational ? (cid:88) (cid:88) (cid:88) (cid:88) –case 4 rational ? (cid:88) (cid:88) – – (cid:88) case 5 rational ? (cid:88) (cid:88) (cid:88) – (cid:88) case 6 rational ? (cid:88) (cid:88) (cid:88) (cid:88) (cid:88) Mathematicians would be interested in case 2 and definitely in case 3. In fact case 3 has been recentlyparameterized[8, 9] (see § Some examples of the smallest integer parallelograms.
The following parallelograms are thesmallest integer parallelograms that exist (the next 2 tables list the parallelograms two to a row).
Table 13.
Case 3. First 10 parallelograms with 2 rational diagonals.sides diags sides diags3 4 5 5 4 7 7 95 5 6 8 5 10 9 135 12 13 13 6 7 7 116 8 10 10 6 13 11 176 17 17 19 7 9 8 14
Table 14.
Case 6. First 10 parallelograms with 2 diagonals rational and area rational.sides diags sides diags3 4 5 5 5 5 6 85 12 13 13 6 8 10 107 24 25 25 8 15 17 179 12 15 15 9 40 41 4110 10 12 16 10 24 26 26Please note that if the 2 diagonals are equal, the parallelogram is a rectangle or square. There are someparallelograms in common with both lists.
LASSIFYING RATIONAL PARALLELEPIPEDS 11
Table 15.
Parallelograms in common - 4 right triangles - 1 Heron triangle.sides diags sides diags3 4 5 5 5 12 13 135 5 6 8 6 8 10 107 24 25 255.4.
Some examples of rational tetrahedrons.
From these smallest integer parallelograms, come thesmallest integer tetrahedrons. A computer search quickly discovers the following integer tetrahedrons.
Table 16.
The first 40 smallest integer tetrahedrons.tetrahedron edges tetrahedrons edges5 6 5 9 10 13 5 6 5 9 10 95 6 5 13 10 13 5 6 5 13 10 95 6 5 13 12 13 5 8 5 9 10 135 8 5 9 10 9 5 8 5 13 10 135 8 5 13 10 9 5 8 5 13 12 137 8 9 21 22 17 7 8 9 21 22 297 8 9 25 22 17 7 8 9 25 22 297 12 11 13 16 15 7 12 11 13 16 237 12 11 21 16 15 7 12 11 21 16 237 13 16 21 22 18 7 13 16 21 22 347 13 16 25 22 18 7 13 16 25 22 347 14 9 21 22 17 7 14 9 21 22 297 14 9 25 22 17 7 14 9 25 22 297 14 11 13 16 15 7 14 11 13 16 237 14 11 21 16 15 7 14 11 21 16 237 21 16 21 22 18 7 21 16 21 22 347 21 16 25 22 18 7 21 16 25 22 347 21 22 25 24 26 7 21 22 25 24 387 25 22 25 24 26 7 25 22 25 24 3813 10 13 85 84 85 13 24 13 85 84 855.5.
The Tetrahedron Permutation Group.
It is important to note that for integer tetrahedrons, usingjust the six side lengths a , b , c , d , e , & f , that up to 24 tetrahedrons can be found in a family. The familyis found by considering all permutations of sides, and accounting for those which create a valid tetrahedron.This permutation group accounts for the appearance of the similar tetrahedrons seen in Table 16 above. Table 17.
The Tetredron Permutation Group a b c d e fa b c (cid:112) a + e ) − d e (cid:112) c + e ) − f a (cid:112) a + c ) − b c d e (cid:112) c + e ) − f a (cid:112) a + c ) − b c (cid:112) a + e ) − d e fa d e b c fa d e b c (cid:112) c + e ) − f a (cid:112) a + e ) − d e (cid:112) a + c ) − b c (cid:112) c + e ) − f continued on next page Table 17 – The Tetradron Permutation Group - continued a (cid:112) a + e ) − d e (cid:112) a + c ) − b c fc b a f e dc b a (cid:112) c + e ) − f e (cid:112) a + e ) − d c (cid:112) a + c ) − b a f e (cid:112) a + e ) − d c (cid:112) a + c ) − b a (cid:112) c + e ) − f e dc f e b a dc f e (cid:112) a + c ) − b a (cid:112) a + e ) − d c (cid:112) c + e ) − f e b a (cid:112) a + e ) − d c (cid:112) c + e ) − f e (cid:112) a + c ) − b a de d a f c be d a (cid:112) c + e ) − f c (cid:112) a + c ) − b e (cid:112) a + e ) − d a f c (cid:112) a + c ) − b e (cid:112) a + e ) − d a (cid:112) c + e ) − f c be f c d a be f c (cid:112) a + e ) − d a (cid:112) a + c ) − b e (cid:112) c + e ) − f c d a (cid:112) a + c ) − b e (cid:112) c + e ) − f c (cid:112) a + e ) − d a b The actual group is a set union of the 24 rows given above.NOTE: While 24 tetrahedrons are usually in the group, sometimes less can occur, due to symmetries ofidentical tetrahedron sides, which can create identical rows. For rectangular pipeds , there are 6 tetrahedronsin the group.5.6.
Density of rational solutions.
The next thing which must be considered is the density of the rationalsolutions in Q as compared to the irrational solutions in R real space for a Diophantine geometrical object,when considering the set of desired rational components. Sometimes this density is very sparse and cleveralgorithms must be utilized to even find the rational solutions that match the desired set.For example, while running computer studies of random parallelepipeds, it became necessary to create allpossible parallelepipeds using 2 integer sides and a 3rd integer diagonal, thus satisfying case 2, and case 5,for the parallelogram, automatically.A record was kept of one such run using 3 integers, a, b, c , two ( a, b ) for the 4 sides and c for 1 rationaldiagonal and ran the integers for 0 < a < = 100 and 0 < b < a and a − b < c < a + b to create theparallelograms. Table 18.
Statistics for a short run of 746,344 integer sided parallelograms.count percent case rational diagonal ? rational area ?737628 98.832% case 2 no no1827 0.2448% case 5 no yes6683 0.8954% case 3 yes no206 0.0276% case 6 yes yes63 0.00844% ibid yes right triangle143 0.01916% ibid yes scalene triangle746344 100.000% – –So it can be seen that rational solutions are sparse, only 206 solutions were found from 746,344 examined,even for a computer algorithm which automatically satisfied four rational sides and 1 rational diagonal ∈ Z . LASSIFYING RATIONAL PARALLELEPIPEDS 13
If we can parameterize the solutions and show that they completely cover the rational solution space in Z or Q , this is a vast improvement in hunting for complete solutions for our set of properties to satisfy.5.7. Parameterizing Heron Triangles and Rational Parallelograms.
In order to efficiently discoverrational solutions to
Diophantine parallelepipeds , it is very convenient to utilize parametric solutions toautomatically speed up the search process efficiently.Such is the case here for both triangles and parallelograms.The Heron Triangle is efficiently parameterized for the computer as the following solution using 4 integerparameters m , n , p , and q , all ∈ Z : A parametric solution[6] for Heron Triangles (6) sides a = mn ( p + q ) b = pq ( m + n ) c = pq ( n − m ) + mn ( q − p ) (cid:52) area = 4 mnpq ( mq + np )( nq − mp )This solution has been proven to fully cover the rational space Q .Another very helpful solution is that for parallelograms with 2 rational diagonals. This has been recentlyparameterized by Walter Wyss[8][9]. I changed his rational solution of u, m, n ∈ Q to that of k, m, n, p, q ∈ Z since the integer case is easier to handle in computers than rational numbers.In this parametric solution, the 5 integer parameters are k , m , n , p , and q . The scaling k parametercreates composite solutions(normally k = 1). The 4 sides of the parallelogram are a, b and the 2 diagonalsare c, d . A parametric solution[9] for rational parallelograms (7) sides a = k ( nq − mp ) b = k ( mq + np )diagonals c = k ( p ( m − n ) + q ( m + n )) d = k ( p ( n + m ) + q ( n − m ))This solution also fully covers rational space Q These 2 parameterization enormously speeded up the recovery of rational triangles and parallelogramsused to assemble rational component parallelepipeds, since the natural construction sequence is triangles → parallelograms → parallelepipeds.5.8. Density of Rational Solutions using Parameterization.
Even with parameterization, the densityof rational solutions still has sparsity. In a run using a parallelogram parameterization[ § < side < ,
001 for the sides, only 0.0867% satisfied the additional rational constraint for case 6 usingthe parameterization.Using the Heron parameterization[ § < sides < ,
001 only 5,302 had the 4th diagonal rational, while 220,221 wereirrational. This meant that only 2.351% satisfied the constraint for case 6. This is better than the previous0.0867% for the parallelogram 2 diagonals rational parameterization, but 2% is still low.In another run of a search program using the Heron parametrization, 72,329,230 Heron triangles werefound but only 704,953 were unique. Of those, 686,264 had one diagonal irrational, only 18,689 satisfied case6 with both diagonals rational and area rational. This means that only 2.651% of the unique solutions werefully rational, but only 0.975% of the initial solutions satisfied case 6 constraints, for a net result of 0.0258%of the raw solutions, even using an efficient Heron triangle parameterization.
From this, it is seen that even when using parameterizations, the sparsity of rational solutions satisfyingconstraints is low. It is necessary to use clever algorithms in computer searching for Diophantine objectsin embedded in 2d space. The requirements are more severe with objects embedded in 3d space, and theseconstraints increase geometrically as we shall see.6.
Computer Searches of Rational Parallelepipeds
We offer some interesting discoveries of pipeds from the computer seaches.To start, the extensive computer runs created 1,981,336,681 unique integer tetrahedrons which requiredover 441.8 gigabytes of storage space to classify. The computer search also found 79,580 degenerate paral-lelepipeds, where the volume = 0, the algorithms creating the pipeds did not specifically check for non-zerovolume while assembling possible pipeds as this would have significantly increased the search time.First of all, the following counts for the five classes of Diophantine parallelepipeds was obtained:class count % abundance acute triclinic .
381 229 379 8 obtuse triclinic .
275 397 680 9 .
340 345 992 9 .
003 024 725 7 rectangular
44 0 .
000 002 220 7
Total count .
000 000 000 0
Table 19.
Counts of Diophantine pipeds in each classThis shows that about 1 obtuse piped occurred for every 2 acute pipeds, while both of them togetherare 292 . × as numerous as pipeds. Both acute and obtuse pipeds together are 32 , . × morenumerous than pipeds, while rectangular pipeds are very scarce, accounting for only 0.0000022207%of the tetrahedrons discovered. Futhermore the ratio between the and pipeds is about 113to 1.There were 33,516 pipeds found which had rational volume, this is 0.0016915852% or only about 1 in59,116 pipeds.6.1.
115 Unique Categories of Diophantine Pipeds.
After sorting the 1,981,336,681 integer tetrahe-drons, it was discovered that 1,923 unique entries existed, grouped by the 27 rationality checks, and 115unique categories resulted.In the category table given below, we refer back to the 27 rationality checks previously determined in § Table 20.
115 Unique categories of Diophantine Parallelepipedscat. count edges skew face body f area b area vol. notes1 268 3 0 6 0 0 0 -12 1914630558 3 0 6 0 0 0 03 14628 3 0 6 0 0 0 1
Continued on next page
LASSIFYING RATIONAL PARALLELEPIPEDS 15
Table 20 –
115 Unique Categories - continued from previous page cat. count edges skew face body f area b area vol. notes4 2585672 3 0 6 0 0 1 05 48 3 0 6 0 0 1 16 13252 3 0 6 0 0 2 07 96 3 0 6 0 0 2 18 48 3 0 6 0 0 3 09 32448 3 0 6 0 1 0 -110 19497256 3 0 6 0 1 0 011 912 3 0 6 0 1 0 112 6831972 3 0 6 0 1 1 013 72 3 0 6 0 1 1 114 1412 3 0 6 0 1 2 015 2304 3 0 6 0 1 2 116 12232 3 0 6 0 1 3 017 69136 3 0 6 0 2 0 018 2368 3 0 6 0 2 1 019 20898 3 0 6 0 2 2 020 110 3 0 6 0 2 4 021 4440 3 0 6 0 3 0 022 19628 3 0 6 0 3 1 023 7962 3 0 6 0 3 2 124 3776 3 0 6 0 3 3 025 44 3 0 6 0 3 6 1 rect
26 24 3 0 6 1 0 0 -127 32314073 3 0 6 1 0 0 028 5036 3 0 6 1 0 0 129 145920 3 0 6 1 0 1 030 192 3 0 6 1 0 1 131 2648 3 0 6 1 0 2 032 5232 3 0 6 1 1 0 -133 477828 3 0 6 1 1 0 034 384 3 0 6 1 1 0 135 65516 3 0 6 1 1 1 036 24 3 0 6 1 1 2 037 208 3 0 6 1 1 2 138 196 3 0 6 1 1 3 039 1892 3 0 6 1 2 0 040 216 3 0 6 1 2 1 041 264 3 0 6 1 3 1 042 834952 3 0 6 2 0 0 043 264 3 0 6 2 0 0 144 24 3 0 6 2 0 1 -145 12240 3 0 6 2 0 1 046 504 3 0 6 2 0 2 047 3056 3 0 6 2 1 0 -148 102768 3 0 6 2 1 0 0
Continued on next page
Table 20 –
115 Unique Categories - continued from previous page cat. count edges skew face body f area b area vol. notes49 53252 3 0 6 2 1 1 050 1320 3 0 6 2 1 3 051 24 3 0 6 2 2 0 052 718 3 0 6 2 2 2 053 72 3 0 6 2 3 1 054 182 3 0 6 2 3 2 155 19416 3 0 6 3 0 0 056 192 3 0 6 3 0 1 057 64 3 0 6 3 1 0 -158 5160 3 0 6 3 1 0 059 1912 3 0 6 3 1 1 060 124 3 0 6 4 0 0 0 perfect
61 600 3 0 6 4 0 3 -162 4 3 0 6 4 1 0 0 perfect
63 1324 3 1 6 0 0 0 -164 3126152 3 1 6 0 0 0 065 96 3 1 6 0 0 0 166 29040 3 1 6 0 0 1 067 648 3 1 6 0 0 2 068 74084 3 1 6 0 1 0 069 336 3 1 6 0 1 0 170 83088 3 1 6 0 1 1 071 144 3 1 6 0 1 1 172 24 3 1 6 0 1 2 073 248 3 1 6 0 1 2 174 300 3 1 6 0 1 3 075 312 3 1 6 0 2 0 076 108 3 1 6 0 3 1 077 120 3 1 6 1 0 0 -178 128176 3 1 6 1 0 0 079 2988 3 1 6 1 0 1 080 5160 3 1 6 1 1 0 081 24 3 1 6 1 1 0 182 2880 3 1 6 1 1 1 083 24 3 1 6 1 1 2 184 72 3 1 6 1 2 0 085 5136 3 1 6 2 0 0 086 48 3 1 6 2 0 0 187 1304 3 1 6 2 0 1 -188 360 3 1 6 2 1 0 089 2460 3 1 6 2 1 1 090 432 3 1 6 2 1 2 091 192 3 1 6 2 1 3 092 72 3 1 6 3 0 0 093 12 3 1 6 3 1 1 0
Continued on next page
LASSIFYING RATIONAL PARALLELEPIPEDS 17
Table 20 –
115 Unique Categories - continued from previous page cat. count edges skew face body f area b area vol. notes94 16352 3 2 6 0 0 0 095 24 3 2 6 0 0 0 196 916 3 2 6 0 0 1 097 1476 3 2 6 0 1 0 098 3388 3 2 6 0 1 1 099 36 3 2 6 0 1 2 1100 12 3 2 6 0 1 3 0101 96 3 2 6 0 2 0 0102 12 3 2 6 0 2 2 0103 36 3 2 6 0 3 1 0104 12 3 2 6 0 3 2 1105 808 3 2 6 1 0 0 0106 3024 3 2 6 1 1 0 0107 1404 3 2 6 1 1 1 0108 192 3 2 6 1 1 2 1109 48 3 2 6 2 1 0 0110 144 3 2 6 2 1 1 0111 180 3 3 6 0 1 1 0112 32744 3 4 6 0 3 6 -1113 48 3 4 6 1 3 6 -1114 2300 3 4 6 2 3 6 -1115 24 3 4 6 4 3 6 -1 planar
We will select interesting examples from these 115 categories of pipeds in the next sections.6.2.
Perfect Diophantine parallelepipeds.
Perfect Diophantine parallelepipeds are those where both the6 face diagonals and the 4 body diagonals are rational.The first perfect parallelepiped discovered was the Diophantine acute piped created by these 3 vectors:vector direction vector length v [106 , ,
0] 106 v [ , √ ,
0] 271 v [ , √ , √ ] 103 Figure 6.
The Sawyer-Reiter Acute Diophantine Piped α = arccos (cid:0) (cid:1) β = arccos (cid:0) (cid:1) γ = arccos (cid:0) (cid:1) and a (cid:54) = b (cid:54) = c This acute piped discovered by Jorge F. Sawyer and Clifford A. Reiter[5] in 2008 or 2009 was the firstknown that had all 4 body diagonals rational as well as all 12 face diagonals rational.The computer searches found 6 perfect Diophantine parallelepipeds including the Sawyer-Reiter perfect pipedshown in Fig. 6. Further computer searches have revealed that the can have all
Table 21.
Six perfect Diophantine parallelepipeds edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
Basis Vectors Tetrahedron SidesClass v v v a b c d e facute (103 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
103 101 106 266 271 255acute (335 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
335 444 365 595 630 3851-ortho (340 , ,
0) (0 , , (cid:16) , , √ (cid:17)
340 493 357 852 952 875acute (342 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
342 463 595 661 739 774acute (375 , , (cid:16) , √ , (cid:17) (cid:16) , − √ , √ (cid:17)
375 285 540 448 647 653acute (422 , , (cid:16) , √ , (cid:17) (cid:16) , − √ , √ (cid:17)
422 431 579 577 925 776 diagonals rational also. Rathbun found parametric formuli for such pipeds[3],[4].
LASSIFYING RATIONAL PARALLELEPIPEDS 19
Interesting examples of Diophantine pipeds found by searches.
In the 14 examples given below,in general, the first occurrences of the designated piped is given.6.3.1.
Rational volume pipeds.
Searching found that Diophantine parallelepipeds can have a rational volume,although none of the body diagonals have a rational length, nor does any of the 6 body or 3 face parallelogramshave a rational area. Interesting enough, the 2 n d and the 3 r d pipeds in Table 22. have the same volume, Table 22.
Rational volume Diophantine parallelepipeds edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1Basis Vectors Tetrahedron SidesClass v v v a b c d e f Volumeacute (17 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
17 32 41 61 72 43 18144obtuse (19 , , (cid:16) , √ , (cid:17) (cid:16) , − √ , √ (cid:17)
19 59 58 22 23 69 20160acute (25 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
25 29 30 45 50 32 20160acute (26 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
26 37 39 46 48 33 30240acute (28 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
28 49 49 55 57 22 28224
Rational pipeds with one body parallelogram area rational and rational volume.
Diophantine pipedswere found that had one body parallelogram with rational area, while the other 5 are irrational. None ofthe face parallelograms had rational area. The volume was rational also.
Table 23.
Rational pipeds with one rational body parallelogram area and rational volume
Basis Vectors Tetrahedron SidesClass v v v a b c d e fobtuse (99 , , (cid:16) , √ , (cid:17) (cid:16) , − √ , √ (cid:17)
99 452 463 220 209 560obtuse (99 , , (cid:16) − , √ , (cid:17) (cid:16) − , − √ , √ (cid:17)
99 494 463 242 209 560acute (274 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
274 298 428 507 617 325obtuse (209 , , (cid:16) , √ , (cid:17) (cid:16) , − √ , √ (cid:17)
209 450 463 220 99 494obtuse (209 , , (cid:16) , √ , (cid:17) (cid:16) − , √ , √ (cid:17)
209 450 463 242 99 452
Table 24.
Rational component checks for the rational body area pipeds edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 11 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 11 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 11 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 11 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1
The volume of these 5 pipeds are 8781696, 8781696, 24710400, 8781696, 8781696. As shown by theseidentical volumes the 1 st , 2 nd , 4 th , and 5 th tetrahedrons are in the same family, while the 3 rd is in anotherfamily. Rational pipeds with all 9 parallelograms of rational area and rational volume.
It was discovered thatthe rectangular class of Diophantine pipeds , has all 3 face parallelograms and all 6 body parallelograms withrational area, as well as the volume rational. Unfortunately none of the 4 body diagonals were rational. Thisclass of piped is related to the question of whether or not perfect cuboids exist, still unanswered.
Table 25.
Parellelepipeds with all face and body parallelograms of rational area edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1Basis Vectors Tetrahedron SidesClass v v v a b c d e f Volumerectangular (44 , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , Rational piped with 1 rational body diagonal, and some rational area parallelograms.
Computer searchesled to the discovery of one piped which had 1 body diagonal rational, and 2 body parallelograms with rationalarea, as well as one face parallelogram with rational area.
Table 26.
Diophantine piped with a rational body diagonal and some parallelograms ofrational area edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0Basis Vectors Tetrahedron SidesClass v v v a b c d e facute (385 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
385 366 415 462 385 432
Rational piped with 2 rational body diagonals and 2 rational area face parallelograms.
One Diophan-tine parallelepiped was found that had 2 rational body diagonals (of 4 possible) and 2 rational area faceparallelograms (of 3 possible).
Table 27.
Diophantine piped with 2 rational body diagonals and 2 rational area faceparallelograms edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 0 1 1 1 1 1 1 0 1 0 1 1 0 1 0 0 0 0 0 0 0Basis Vectors Tetrahedron SidesClass v v v a b c d e f1-ortho (175 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
175 296 255 455 420 375
LASSIFYING RATIONAL PARALLELEPIPEDS 21
Rational piped with 4 rational body diagonals and 1 rational area face parallelogram.
One Diophantinepiped was found that had all 4 body diagonals rational, and had one face parallelogram with rationalarea.
Table 28.
Diophantine piped with 4 rational body diagonals and 1 rational area faceparallelogram edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0Basis Vectors Tetrahedron SidesClass v v v a b c d e f1-ortho (340 , ,
0) (0 , , (cid:16) , , √ (cid:17)
340 493 357 852 952 875
Rational piped with 1 rational area face and 2 rational area body parallelograms.
One piped was foundthat had a rational face area parallelogram and 2 rational area body parallelograms.
Table 29.
Diophantine piped with 1 rational area face and 2 rational area body parallelograms edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 0Basis Vectors Tetrahedron SidesClass v v v a b c d e f1-ortho (204 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
204 204 240 629 595 595
Rational piped with 1 rational body diagonal, 1 rational face parallelogram and 2 rational body paral-lelograms and rational volume.
One Diophantine piped was found which has 1 rational body diagonal, 1 faceparallelogram with rational area, and 2 body parallelograms with rational area, and rational volume.
Table 30.
Diophantine piped with 1 rational body diagonal and rational body/face paral-lelograms edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 0 1 0 0 1 1
Basis Vectors Tetrahedron SidesClass v v v a b c d e f Volumeacute (137 , , (cid:0) , , (cid:1) (cid:0) , , (cid:1)
137 176 137 250 291 250 3991680
Rational piped with 3 rational body diagonals, 1 rational face parallelogram and 1 rational body parallel-ogram.
One parallelepiped was found that had 3 rational body diagonals, 1 rational area face parallelogramand 1 rational area body parallelogram.
Table 31.
Diophantine piped with 3 rational body diagonals and 1 rational face and 1rational body parallelograms edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 1 0 0 1 1 1 1 1 1 1 0 1 1 0 0 1 0 0 0 1 0 0 0
Basis Vectors Tetrahedron SidesClass v v v a b c d e facute (273 , , (cid:16) , √ , (cid:17) (cid:16) , √ , √ (cid:17)
273 431 442 431 442 560
Rational piped with 1 rational area face parallelogram and 3 rational area body parallelograms.
Com-puter searching found a Diophantine piped which had 1 face parallelogram with rational area, and 3 bodyparallelograms with rational area.
Table 32.
Rational piped with 1 rational area face parallelogram and 3 rational area bodyparallelograms edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0
Basis Vectors Tetrahedron SidesClass v v v a b c d e fobtuse (361 , , (cid:16) − , √ , (cid:17) (cid:16) , √ , √ (cid:17)
361 663 424 425 424 448
Rational pipeds with 3 rational face parallelograms and 2 rational body parallelograms.
Computersearching found two examples of pipeds, which have all the face parallelograms rational and 2body parallelograms rational (out of 4). These pipeds are closely related to the face cuboids found in therational cuboid table solutions.
Table 33.
Diophantine cuboids related to face cuboids edges skew triangles face diagonals body diagonals face area body area vol1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 0 0 11 1 1 0 1 1 0 1 1 1 1 1 1 0 0 0 0 1 1 1 0 0 1 0 0 1 1Basis Vectors Tetrahedron SidesClass v v v a b c d e f Volume2-ortho (153 , ,
0) (0 , , (cid:0) , , (cid:1)
153 697 680 697 680 208 213857282-ortho (185 , , (cid:0) , , (cid:1) (0 , , LASSIFYING RATIONAL PARALLELEPIPEDS 23
Rational pipeds with 2 rational area skew triangles and 2 rational body diagonals and 1 rationalarea face parallelogram and 1 rational body parallelogram.
The computer searches found some interestingDiophantine pipeds which had 2 skew triangles with rational area and 2 rational body diagonals, with 1 faceparallelogram rational and 1 body parallelogram rational.Pipeds with rational skew area triangles were quite infrequent.
Table 34.
Diophantine pipeds with 2 rational area skew triangles edges skew triangles face diagonals body diagonals face area body area vol1 1 1 1 0 1 0 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 0 0 01 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1 0 0 1 0 0 0 1 0 0 01 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 0 0 0 1 0 0 0 0 0
Basis Vectors Tetrahedron SidesClass v v v a b c d e fobtuse (36 , , (cid:16) , √ , (cid:17) (cid:16) , − √ , √ (cid:17)
36 95 89 95 89 160acute (96 , , (cid:16) , √ , (cid:17) (cid:16) , − √ , √ (cid:17)
96 175 203 175 203 280obtuse (593 , , (cid:16) , √ , (cid:17) (cid:16) , − √ , √ (cid:17)
593 879 736 736 593 1007
Closest rational Diophantine (degenerate) piped to a superperfect piped.
The computer searches foundsome interesting pipeds in which all the rational checks were rational, except the volume was was 0, indicatingthat these pipeds were degenerate planar figures.What is interesting is that these are . Table 35.
Diophantine piped with all rational components except 0 volume edges skew triangles face diagonals body diagonals face area body area vol1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1
Basis Vectors Tetrahedron SidesClass v v v a b c d e f2-ortho (120 , ,
0) (0 , ,
0) (0 , ,
0) 120 218 182 241 209 27
Rectangular Diophantine pipeds.
There were 46 rectangular
Diophantine pipeds found during thecomputer searches. These are the body type of cuboid solutions. Unfortunately none of them has a rationalbody diagonal leading to a perfect cuboid.These pipeds all have the same signature for their rationality checks.
Table 36.
Rational Component checks for the
Rectangular
Diophantine pipeds edges skew triangles face diagonals body diagonals face area body area vol1 1 1 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1
Table 37.
Rectangular
Diophantine ParallelepipedsClass v v v a b c d e frectangular (44 , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , ,
44) 117 267 240 125 44 244rectangular (132 , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , ,
85) 132 732 720 157 85 725rectangular (132 , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , ,
88) 234 534 480 250 88 488rectangular (240 , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , ,
44) 240 267 117 244 44 125rectangular (240 , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , Continued on next page
LASSIFYING RATIONAL PARALLELEPIPEDS 25
Table 37 –
Rectangular Pipeds - continued from previous page
Class v v v a b c d e frectangular (252 , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , ,
88) 480 534 234 488 88 250rectangular (480 , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , ,
85) 720 732 132 725 85 157rectangular (720 , ,
0) (0 , ,
0) (0 , , , ,
0) (0 , ,
0) (0 , , The Rectangular Diophantine Piped and The Perfect Cuboid
The most commonly known Diophantine parallelepiped known today is the rectangular piped , often called”the brick” or ”box” or ”Euler brick” also known as the cuboid or integer cuboid in math literature. Thereare 7 lengths of interest in this solid, the 3 edges, the 3 face diagonals and the body diagonal.It has been found by search, that there are 3 cases which exist, called the ”body cuboid”, the ”face cuboid”,and the ”edge cuboid”; where the 7th length is irrational. This is problem D18 in Unsolved Problems inNumber Theory (UPINT) by Richard K. Guy[2].Walter Wyss recently attempted to show that the 7th value cannot be rational by utilizing the proof ofLeonard Euler that the sum or difference of 2 quartics cannot be a square[10]. He was able to parametrizethe monoclinic piped and showed that approaching a right angle sets up the impossible quartic conditionwhich Euler disproved. But he failed to prove that no perfect integer cuboid exists, hence question is stillopen today.
Recently Renyxa D’Arox recently reached a milestone[1] with computer searching extending to 2 =9007199254740992 for the body diagonal, if a perfect cuboid exists, its body diagonal must exceed this value.8. Possible Future Studies for Diophantine Analysis
There are some conjectures and questions created by the computer study of the 1,981,336,681 Diophantinepipeds.Some questions: • Is there a rational piped with 4 rational area skew triangles? • Is there a perfect parallelepiped with 2 or 3 rational area face parallelograms? • Is there any perfect parallelepiped with 1..6 rational area body parallelograms? • Is there any perfect parallelepipeds with a rational area skew triangle? 2? 3? or 4? • Is there a perfect parallelepiped with rational volume? (asked by Sawyer, Reiter)[5] • Is there a perfect parallelepiped on the rational lattice? (asked by Sawyer, Reiter)[5]
Two conjectures: • Conjecture 1. Other Diophantine pipeds do exist with less than 6 rational face diagonals as coveredby the computer studies done here • Conjecture 2. No Diophantine pipeds exist with all 27 components rational
These are just some questions and two conjectures relating to the computer search results.Some interesting problems for further analysis are the following: • Parametrize the acute or obtuse triclinic piped as an infiinite family • Parametrize the mono-orthogonal or biclinic piped as an infinite family Appendix A - Proof of 5 Parallelepiped Classes
Let the parallelepiped be situated so the the main vertice v determined by the 3 basis vectors is at theorigin [0,0,0]. Figure 7 shows the parallelepiped and its 8 labeled vertices. Previously, Figure 1 showed the3 basis vectors, the same piped is shown. v v v v v v v v Figure 7.
Labeled vertices of the parallelepiped.The origin [0,0,0] is labeled v . The (cid:126)a basis vector lies along the positive x-axis, v to v ; the (cid:126)b basis vectoris in the x-y plane, along v to v ; and the (cid:126)c basis vector is along the line from v to v , with a non-zero zcomponent.We label the surface angles, as found by the angle traced by the 3 ordered vertices. LASSIFYING RATIONAL PARALLELEPIPEDS 27
Table 38.
The 3 surface angles at each vertice of parallelepipedvertice angle angle angle v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v We know that the interior angles on the diagonals of a parallelogram are equal, and that 2 adjacent anglesare supplementary. This fact enables us to substitute the equivalent angles as necessary for the piped, inorder to redefine all surface angles as a subset of the 3 surface angles at vertice a − b plane a − c plane b − c plane v v v v v v v v v v v v = v v v v v v = 180 ◦ − v v v v v v = v v v v v v = 180 ◦ − v v v v v v = 180 ◦ − v v v v v v = 180 ◦ − v v v v v v = 180 ◦ − v v v v v v = v v v v v v = 180 ◦ − v v v v v v = v v v v v v = v v v v v v = v v v v v v = v v v v v v = 180 ◦ − v v v v v v = v v v v v v = 180 ◦ − v v v v v v = 180 ◦ − v v v v v v = 180 ◦ − v v v v v v = 180 ◦ − v v v v v v = v v v v v v = 180 ◦ − v v v We substitute the 3 surface angles or their supplements found at vertice angle angle angle v v v v v v v v v ◦ − v v v ◦ − v v v v v v ◦ − v v v ◦ − v v v v v v v v v ◦ − v v v ◦ − v v v ◦ − v v v ◦ − v v v v v v v v v ◦ − v v v ◦ − v v v v v v ◦ − v v v ◦ − v v v v v v v v v v v v In order to classify the angles, we consider the such that0 ◦ < < ◦ → sign (cos( )) = 1= 90 ◦ → sign (cos( )) = 0180 ◦ > > ◦ → sign (cos( )) = − ° , 90 ° , and 60 ° respectively. We notice that if the sign of an angle = 1, then its supplement = -1, or vice versa. If the sign of an angle= 0, then the supplement is also = 0.Since all other 7 vertices are given as some combination of the 3 surface angles of vertice
Table 39.
The set union set of the 27 permutations for all 8 vertices of the parallelepiped
We discover that 5 unique classes result, after creating a set union of the 27 permutation set union groupsshown in Table 39.
Table 40.
Five classes of parallelepipedsname classrectangular [0, 0, 0]monoclinic [-1, 0, 0], [0, 0, 1]bi-clinic [-1, -1, 0], [-1, 0, 1], [0, 1, 1]obtuse triclinic [-1, -1, -1], [-1, 1, 1]acute triclinic [-1, -1, 1], [1, 1, 1] (cid:3)
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Some new parameterizations for the Diophantine bi-orthogonal monoclinic piped ,https://arxiv.org/abs/1709.01138 version [v1] Mon, 4 Sep 2017 20:02:15 UTC[4] Randall Rathbun,
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